Summary: METRIC TENSOR ESTIMATES, GEOMETRIC
CONVERGENCE, AND INVERSE BOUNDARY PROBLEMS
Michael Anderson, Atsushi Katsuda, Yaroslav
Kurylev, Matti Lassas, and Michael Taylor
Abstract. Three themes are treated in the results announced here. The rst is the
regularity of a metric tensor, on a manifold with boundary, on which there are given
Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound
on the mean curvature of the boundary. The second is the geometric convergence
of a (sub)sequence of manifolds with boundary with such geometrical bounds and
also an upper bound on the diameter and a lower bound on injectivity and boundary
injectivity radius, making use of the rst part. The third theme involves the unique-
ness and conditional stability of an inverse problem proposed by Gel'fand, making
essential use of the results of the rst two parts.
1. Introduction
Here we announce results on regularity, up to the boundary, of the metric tensor
of a Riemannian manifold with boundary, under Ricci curvature bounds and control
of the boundary's mean curvature; an application to results on Gromov compactness
and geometric convergence in the category of manifolds with boundary; and then
an application of these results to the study of an inverse boundary spectral problem
introduced by I. Gel'fand. Details are given in [AK2LT].